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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>The variation of temperature in the bar is governed by a partial differential equation. The equation is called the <dfn class="terminology">heat conduction equation</dfn>, and has the form</p>
<div class="displaymath process-math" data-contains-math-knowls="  ">
\begin{equation*}
u_t=\alpha^2u_{xx}\label{heatpde},\quad 0&lt;x&lt;L,\quad t&gt;0,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\alpha^2\)</span> is a constant known as the thermal diffusivity. In addition, we assume that the initial temperature distribution in the bar is given; thus</p>
<div class="displaymath process-math" data-contains-math-knowls="  ">
\begin{equation*}
u(x,0)=f(x),\quad 0\leq x\leq L,\label{heatic}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(f\)</span> is a given function. Finally, we assume that the ends of the bar are held at fixed temperature zero:</p>
<div class="displaymath process-math" data-contains-math-knowls="  ">
\begin{equation*}
u(0,t)=0,\quad u(L,t)=0,\quad t&gt;0.\label{heatbc}
\end{equation*}
</div>
<p class="continuation">The fundamental problem of heat conduction is to find <span class="process-math">\(u(x, t)\)</span> that satisfies the <dfn class="terminology">boundary-value problem</dfn> (BVP), that is the partial differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> together with the initial condition <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatic" missing or not unique]</code> and boundary conditions <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code>.</p>
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